 Rational functions with prescribed poles

For finitely many distinct points $P_i$ on a smooth projective genus-$g$ curve, choose $N\ge 2g+1$. The Riemann-Roch theorem gives $\ell (N{P}_i)-\ell ((N-1)P_i)=1$, so choose $f_i$ with a sole pole of exact order $N$ at $P_i$. Then ${\sum }_i{f}_i$ has exactly those poles with that order: all other summands are regular at each $P_i$, so no principal part can cancel. This is an explicit construction rather than an unspecified generic choice of section.